Page 162 - Applied statistics and probability for engineers

140 Chapter 4/Continuous Random Variables and Probability Distributions

Because P X > x) = − ( )
(
1
F x , the probability density function of X equals the negative of the
derivative of the right-hand side of the previous equation. After extensive algebraic simpliica-
tion, the probability density function of X can be shown to equal
x e
f x ( ) = λ r r−1 − x λ for x > 0 and r = 1, 2, . . . .
( r − )!1
This probability density function deines an Erlang random variable. Clearly, an Erlang
random variable with r = 1 is an exponential random variable.
It is convenient to generalize the Erlang distribution to allow r to assume any non-negative
value. Then the Erlang and some other common distributions become special cases of this
(
generalized distribution. To accomplish this step, the factorial function r − ) 1 ! is generalized
(
to apply to any non-negative value of r, but the generalized function should still equal r − ) 1 !
when r is a positive integer.
Gamma Function
The gamma function is
∞
∫
r
−1
r
Γ( ) = x e −x dx, for r > 0 (4-18)
0
It can be shown that the integral in the deinition of Γ( ) r is inite. Furthermore, by using inte-
gration by parts, it can be shown that
r
Γ( ) = (r − )Γ(r − ) 1
1
This result is left as an exercise. Therefore, if r is a positive integer (as in the Erlang distribution),
r
Γ( ) = (r − )!
1
/
1 2
0
Also, Γ( ) = ! = 1 and it can be shown that Γ(1 2 ) = π . The gamma function can be inter-
1
/
preted as a generalization to noninteger values of r of the term that is used in the Erlang prob-
ability density function. Now the Erlang distribution can be generalized.
Gamma Distribution
The random variable X with probability density function
x e
f x ( ) = λ r r−1 r) − x λ , for x > 0 (4-19)
Γ(
is a gamma random variable with parameters λ > 0 and r > 0. If r is an integer, X
has an Erlang distribution.

The parameters λ and r are often called the scale and shape parameters, respectively. How-
ever, one should check the deinitions used in software packages. For example, some sta-
tistical software deines the scale parameter as 1/ λ. Sketches of the gamma distribution for
several values of λ and r are shown in Fig. 4-25. Many different shapes can be generated from
changes to the parameters. Also, the change of variable u = λ x and the deinition of the gamma
function can be used to show that the probability density function integrates to 1.
For the special case when r is an integer and the value of r is not large, Equation (4-17)
can be applied to calculate probabilities for a gamma random variable. However, in general,
the integral of the gamma probability density function is dificult to evaluate so computer
software is used to determine probabilities.
Recall that for an exponential distribution with parameter λ, the mean and variance are 1/ λ
2
and 1/ λ , respectively. An Erlang random variable is the time until the rth event in a Poisson
process and the time between events are independent. Therefore, it is plausible that the mean